All Questions
Tagged with co.combinatorics mg.metric-geometry
246 questions
0
votes
1
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115
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Invariance of Minkowski sum of sets
Given an euclidean space $E$, two sets $A,B\subset E$ and the action on $E$ of two groups $G_A,G_B$ such that $G_A A=A$ and $G_B B=B$, it is possible to generate a group that leaves invariant $A\oplus ...
- 45k
1
vote
0
answers
154
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Volume of a polytope as its degenerates to be lower dimensional
Consider a polytope $P$ defined by the usual inequalities $A\mathbf{x}\leq \mathbf{b}$; let me assume that $P$ is not contained in a proper subspace. A result which I believe to true, but am not ...
- 44.7k
4
votes
2
answers
150
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$O(n^{2-\epsilon})$ bound on choosing $n$ points on the hypersphere to maximize $\pm 1$ weighted sum of their $\binom{n}{2}$ inner products
Given $n,d\in\mathbb{N}, n\gg d$, I'm looking for a bound on the maximum (or minimum) expected value of the following game:
Draw a vector $\epsilon\in\{\pm 1\}^{\binom{n}{2}}$, uniformly at random. ...
- 109k
4
votes
1
answer
322
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Combinatorics related plane geometry
There are $n$ men, standing one at each vertex of a convex $n$-gon. If they are allowed to move together along sides or diagonals of the polygon to reach another vertex, how many different ways are ...
- 82.7k
0
votes
0
answers
64
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Convexity of stars in a subdivision of a simplex
Are the stars of vertices in a barycentric subdivision of a (standard) n-simplex convex?
I assume that it must be valid, but concretely to prove it I come again and again to obstacles. Approaches so ...
- 1
1
vote
1
answer
616
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Polynomial invariant — from product formula to monomial expansion
Context
This question deals with the polynomial invariant denoted by $ H_{n} $ in Maksym Fedorchuk and Igor Pak's 2004 paper Rigidity and polynomial invariants of convex polytopes (sections 7.6 and 9)....
- 149
20
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5
answers
1k
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From convex polytopes to toric varieties: the constructions of Davis and Januszkiewicz
One of the most useful tools in the study of convex polytopes is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on ...
- 63.9k
9
votes
1
answer
200
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Bi-partitioning $2n$ points on the plane with a straight line
Let $S$ be a set of $2n$ points in $\mathbb{R}^2$. Which is the maximum number of different bi-partitions of $S$ generated by a straight line?
More precisely, which is the maximum number of partitions ...
- 18.8k
21
votes
3
answers
935
views
Cutting of a regular polygon into congruent pieces
Question. For which $N$ it is possible to cut a regular $N$-gon into congruent pieces such that the center of the regular polygon lies strictly inside one of the pieces? For $N=3,4$ there are trivial ...
- 691
2
votes
1
answer
84
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What is the average component size of a coloring?
Supose each cell of a big (or infinite) grid is colored at random by one of $k$ colors. Then the connected monochromatic components (here components are not supposed to contain "wasp waists",...
- 4,589
9
votes
3
answers
1k
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Generalization of Sylvester-Gallai theorem
The Sylvester-Gallai theorem states that it is not possible to arrange a finite number
of points so that a line through every two of them passes through a
third unless they are all on a single ...
- 32.1k
5
votes
1
answer
190
views
Number of distinct normalized vectors from the center of a hexagon in a hexagonal grid
Consider an infinite hexagonal grid composed of regular hexagons. Choose any hex to be the origin hex. Let n be a natural number.
Find an expression, in terms of n, for the number of distinct ...
- 60.5k
3
votes
0
answers
59
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2-complexes which are coarse-grained graphs
A polygonal complex $K$ is said to be geometrically 2-dimensional if the topological space it defines is a surface (boundaries are allowed). It is said to be $C$-quasi-1-dimensional (for some $C>0$)...
- 63.9k
8
votes
3
answers
390
views
Is there any edge- but not vertex-transitive polytope in $d\ge 4$ dimensions?
I consider convex polytopes $P\subset\Bbb R^d$. The polytope is called vertex- resp. edge-transitive, if any vertex resp. edge can be mapped to any other by a symmetry of the polytope.
I am looking ...
- 13.6k
7
votes
0
answers
254
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Set of unit vectors such that among any three there is an orthogonal pair
I was fascinated by the solutions of Problem 8 of the IMC 2021 contest, which can be summarized as:
Theorem 1. Let $v_1,\dotsc,v_N$ be distinct unit vectors in $\mathbb{R}^n$ such that among any three ...
- 105k
3
votes
1
answer
179
views
When is a $k$-distance-transitive graph already distance-transitive?
Call a (finite and connected) graph $k$-distance-transitive if its symmetry group acts transitively on the pairs in each one of the sets
$$D_\delta:=\{(i,j)\in V\times V\mid \mathrm d(i,j)=\delta\},\...
CommunityBot
- 1
28
votes
5
answers
2k
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Visibility of vertices in polyhedra
Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ is an interior point of $P$. Is it true that $X$ can see at least one vertex of $P$? More ...
- 32.1k
5
votes
0
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235
views
Arrangement of points, lines, and planes
Is it possible to construct a finite nontrivial arrangement of points, lines, and planes in 3-dimensional Euclidean space with the following properties?
every line is incident with four points and ...
- 1,444
3
votes
1
answer
206
views
Random planes separating points in $\mathbb{R}^3$
We are given a unit origin-centered sphere $S$ in $\mathbb{R}^3$, and three points $\mathbf{x},\mathbf{y},\mathbf{z}\in S$. Let $\mathbf{h}$ be a point selected uniformly at random from $S$ and let $H$...
- 128k
2
votes
0
answers
131
views
Maximum number of regions in a disk partitioned by pairs of parallel chords
We are given a disk $D$ in $\mathbb{R}^2$. Let $C$ be its boundary (i.e., the circle bounding $D$ on its plane). Let $P(n,d)$ be a set of $n$ pairs of chords of $C$ such that for each $\{c,c'\}\in P(n,...
- 1,677
5
votes
0
answers
190
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The existence of $n$-sided cells in regular $m$-gons
For any integer $n >= 3$, does there exist a regular
$m$-gon with all diagonals drawn containing a cell with $n$ sides?
See A342222 and its cross-references.
Regular polygon on the Wiki.
&...
- 251
2
votes
1
answer
151
views
Given an input point in $\mathbb{R}^n$, select (one of) the closest point(s) from a fixed large set of points given in advance
We are given a set $S$ of $m\gg 1$ points in $\mathbb{R}^n$.
In the problem I am trying to solve, in a sequential fashion, we obtain a new point $p_r\not\in S$ at each round $r\ge 1$ and the goal is ...
- 151k
2
votes
1
answer
143
views
Triangles and convex hulls in high dimensions
Given a set $S_n$ of $n$ points $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n\in\mathbb{R}^d$, such that every $(d+1)$-tuple in $S_n$ is affinely independent, and let $C(S_n)$ be the convex hull ...
- 151k
1
vote
0
answers
124
views
Number of lattice points in a structural symmetric convex body
Let $f$ is a convex symmetric function on the interval $[-a,a]$, i.e., $f(-x)=f(x)$ for $\forall \, x\in [-a,a]$. Then we consider a $n$-dimensional convex body in Euclidean space
\begin{equation}
\...
- 13.9k
22
votes
1
answer
886
views
Happy ants never leave compact domain?
I am curious if the following seemingly simple question has an easy answer?
Consider an ant population of $N$ ants that lives in $\mathbb R^2$. Each ant can be labeled by some coordinate $x\in \mathbb ...
- 148k
2
votes
1
answer
151
views
Graph immersed into the plane with segments as edges and we search for matching with no edges intersecting
There are some points in the plane and some of them are connected with segments between them. We look at this structure as a graph immersed into the plane where the points are the vertices and the ...
- 12.9k
3
votes
1
answer
143
views
Combinatorial Euclidean geometry problem
Let $\mathcal{S}^d_{\epsilon}$ be the collection of all sets $S:=\{\mathbf{x}_1, \mathbf{x}_2, \ldots \mathbf{x}_{d+1}\}$ of $d+1$ points in a $d$-dimensional Euclidean space such that, for a given ...
- 6,101
1
vote
1
answer
347
views
Upper bounds for high-dimensional spherical codes given the covering radius
I assume that this sort of question has already been considered at great length. Nevertheless, I could not find an answer to this question in the related literature.
Given a constant $a\in (0,2]$, ...
- 10.4k
2
votes
1
answer
404
views
Euclidean distance bound with geometric constraints
Let $S_n$ be a set of $n$ points belonging to $\mathcal{B}_d:=\{\mathbf{x}\in\mathbb{R}^d:\|\mathbf{x}\|_2\le 1\}$, where $d\ll \log(n)$.
Let $s_n$ and $\ell_n$ be respectively defined as follows:
$$...
- 1,677
6
votes
1
answer
424
views
Probability of intersecting a rectangle with random straight lines
We are given a rectangle $R$ with sides lengths $r_1$ and $r_2$, contained in a square $S$, with sides lengths $s_1=s_2\ge r_1$ and $s_2=s_1\ge r_2$. $R$ and $S$ are axis-aligned in a cartesian plane $...
- 1,677
2
votes
2
answers
379
views
Probabilistic combinatorial optimization problem on the distances between pairs of points in $[0,1]$
Let $S$ be a set of $n \gg 1$ points lying on the interval $[0,1]$. Given a point $p\in[0,1]$, let $S_p\subseteq S\times S$ be the set formed by all pairs of points $(x,y)$ with $x,y\in S$, such that ...
- 111
4
votes
0
answers
92
views
Possible cardinalities of spherical tiling
Suppose that we have a tiling of $n$-dimensional (I want to get answer for $n = 4$, but general result would be nice!) sphere by isometric tiles strictly contained inside the right-angled simplex. ...
- 13.6k
9
votes
1
answer
160
views
Hyperplane arrangements whose regions all have the same shape
Suppose I have a (finite, real, central, essential) hyperplane arrangement $\mathcal{H}$ such that all regions "have the same shape": for any two regions $R,R'$, there is an orthogonal ...
- 3,271
2
votes
0
answers
131
views
Optimal way to group points in the plane into clusters
Consider a strictly decreasing sequence $d = (d_k)_{k\ge 1}$ of distances in $(0,1)$. Given a constant $C>2$, we say that $d$ has the $C$-grouping property if any finite non-empty subset $S$ (of ...
- 1,761
1
vote
0
answers
81
views
Constructive way to optimally cover a compact subset of Euclidean space
Let, $(X,d)$ be a simply connected compact subset of $\mathbb{R}^d$ with non-empty interiorn, let $d$ denote the Euclidean metric, and let $\varepsilon>0$. Is there a way to iteratively select ...
- 5,405
17
votes
4
answers
1k
views
Can I build infinitely many polytopes from only finitely many prescribed facets?
Given a finite set of convex $d$-dimensional polytopes $\mathcal P$, for some $d\ge 2$.
Question: Is it true that there are only finitely many different convex $(d+1)$-dimensional polytopes whose ...
- 4,713
5
votes
1
answer
114
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Packing in uniform domains
Given $N$ points $X:=(x_i)_{i \in \{1,..,N\}}$, we now define a score function $S:X \rightarrow \mathbb{N}$ that is $S(X)= \sum_{i=1}^N S(x_i)$ where the score of $S(x_i)$ is
$$S(x_i) = 2* \vert \{x_j;...
- 28.9k
5
votes
0
answers
313
views
Trade-off between covering number, ball radius and diameter of $d$-dimensional shapes
Given any $d$-dimensional shape $X$ in the Euclidean space, let $\ell(X)$ be the length of the longest line segment connecting two points of $X$. How can we prove the following statement?
There exists ...
- 1,677
4
votes
1
answer
363
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Trade-off between hypervolume and diameter of $d$-dimensional shapes having a hypercubic smallest bounding box
Given any $d$-dimensional shape $X$, let $V(X)$ be its $d$-dimensional volume, and let $\ell(X)$ be the length of the longest line segment connecting two points of $X$.
Let $\mathcal{S}_C$ be the set ...
- 1,677
3
votes
0
answers
134
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Two questions on counterexamples to Borsuk's conjecture and ball-packings
In 1933 Karol Borsuk conjectured the following
Can every bounded subset $E$ of $\mathbb{R}^d$ be partitioned into $(d+1)$ sets, each of which has a smaller diameter than $E$?
Whilst new to this ...
- 63.9k
5
votes
0
answers
199
views
Existence of a honeycomb composed by nearly-hyperspherical $d$-dimensional cells having the same shape and size
Let $\mathcal{H}$ the class of all honeycombs composed by $d$-dimensional cells $C$ having all the same shape and size in a $d$-dimensional space $\mathcal{S}$.
Let $s(C)$ and $\ell(C)$ be ...
- 1,677
7
votes
3
answers
1k
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How can we count lines in an n-x-n rectangular array?
Is there a formula for the number of lines that contain exactly two points through an n x n rectangular array of points?
- 4,713
2
votes
3
answers
290
views
Geometric probabilistic problem on triangles on a plane
We are given a triangle $T$ on a plane $P$, with sidelengths $a$, $b$ and $c$, where $c \ge b \ge a > 0$. A straight line $L$ on $P$ is selected uniformly at random from the set of all the ...
- 61.9k
4
votes
0
answers
246
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Distance properties of the permutations of a set of points in a Euclidean space
We are given a set of $n$ distinct points $S=\{\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n\}$ in a Euclidean space $\mathbb{R}^d$, where the distance between two points $\mathbf{x}_i,\mathbf{x}_j\...
- 1,677
5
votes
2
answers
304
views
Is there a 4-polytope without 3-gonal and 4-gonal faces, other than the 120-cell?
The question is in the title:
Question: Is there any 4-dimensional polytope without 3-gonal and 4-gonal faces (of dimension two), other than the 120-cell?
I consider only convex polytopes (convex ...
- 13.6k
13
votes
2
answers
3k
views
How many squares can be formed by using n points?
How many squares can be formed by using n points on a 3 dimensional space?
Like using 4 points, there is 1 square be formed
Using 5 points, still 1 square
Using 6 points, 3 squares can be formed
- 13k
5
votes
1
answer
361
views
What is known about the duals of cyclic polytopes?
What is known about the duals of cyclic polytopes, in particular, their facets (or equivalently, the vertex-figures of cyclic polytopes)?
In even dimensions, all facets of the dual are ...
4
votes
1
answer
293
views
Number of points in a lattice and an oblong box
I have a very simple question in geometry of numbers. (It is a slight modification of Counting points on the intersection of a box and a lattice .) There's a bound I can easily prove, and it's good ...
CommunityBot
- 1
8
votes
1
answer
270
views
Sizes of connected components from a random choice in a grid
This is inspired by the illustration in this recently updated question. So we take a (fairly big) $n$ and an $n \times n$ grid where we draw at random one diagonal in each of the $1 \times 1$ squares. ...
- 7,514
14
votes
2
answers
878
views
Sets of evenly distributed points in the Euclidean plane
Is there a set $P \subset \mathbb{R}^2$ of points in the Euclidean plane whose intersection
with every convex subset of $\mathbb{R}^2$ of area $1$ is nonempty but finite?
If the answer is yes, can $P$...
- 18.8k